In a medium bowl, lightly toss the crabmeat with the cracker crumbs. Gently fold in the mayonnaise mixture. You want the mayonnaise mixture to incorporate into the crabmeat, but be careful not to overmix. I like to use my hands, or a spatula for the best results. Cover and refrigerate for at least 1 hour. You can also refrigerate overnight.
Using a 1/3 cup measuring cup, scoop the crab mixture into 8 separate mounds. Very lightly pack into 8 patties, about 1 1/2 inches thick. Don’t pack down too much – you don’t want flat crab cakes. They should be slightly rounded on top.
In a large skillet, heat the oil until shimmering. Add the crab cakes and cook over moderately high heat until deeply golden and heated through, about 3 minutes per side.
Transfer the crab cakes to plates and serve with lemon wedges.